Risk and Returns: Geometric Average Returns

Kerry Back

BUSI 721, Fall 2022
JGSB, Rice University

Tesla went down 50% between Nov 2021 and May 2022.

It then went up 50% between May 2022 and Aug 2022.

Were Tesla shareholders back to even?

For each $100 of Tesla stock, shareholders experienced

100 → 50 → 75

They lost 25%, even though the arithmetic average return was zero.

So, lose 50% and make 50% → lose 25%. Suppose you
- make 50% and then lose 50%?
- lose 50% and then make 100%?
- make 100% and then lose 50%?

Geometric Average Return

Given returns $r_1, \cdots, r_n,$
the geometric average return is the number r such that

$(1+r)^{n}=(1+r_1)\cdots(1+r_{n})$

So earning r each period produces the same accumulation as the actual returns $r_1, \cdots, r_n.$

We solve for it as

\[r=[(1+r_1)\cdots(1+r_n)]^{1/n}-1\]

The geometric average return is always less than the arithmetic average return.

Examples

make 50% and lose 50% → geometric average is \[\sqrt{1.5 x 0.5}-1=-0.134\]

make 100% and lose 50% → geometric average is \[\sqrt{2 x 0.5}-1=0\]